Linear and geometrically nonlinear piecewise hierarchicalp-version curved beam elements for laminated composites

A piecewise hierarchical displacement based p-version two and three dimensional curved beam finite elements have been developed for linear static and geometrically nonlinear static behavior for laminated composites. The element geometry is defined by the nodes located at the center line of the element and by the nodal width and thickness vectors. The individual lamina stiffness matrices and the equivalent nodal force vectors are derived using the principle of virtual work and the hierarchical displacements approximation functions for each lamina of the laminate. Interlamina continuity conditions of displacements at the interlamina boundaries are imposed and conveniently arranged in the form of a transformation matrix which allow the transformation of the lamina degrees of freedom to the laminate degrees of freedom. These transformation matrices are used to transform the lamina stiffness matrices and equivalent load vectors to the laminate stiffness matrix and the equivalent nodal force vector. The p-version lamina displacement approximations are derived directly from the Lagrange interpolating polynomials. The element approximation functions ensure C{dollar}sp0{dollar} continuity of displacements across the interelement and interlamina boundaries. Incremental equations for geometrically nonlinear static behavior are derived and solved using the standard Newton's method. Numerical examples are presented to demonstrate the accuracy, convergence characteristics, and the ability of the formulation to handle smooth as well as singular problems in laminated composites. The formulation yields interlamina continuity and/or discontinuity of the stresses and strains which are in perfect agreement with the physics of laminate behavior. Numerical results are compared with the analytical solutions and existing formulations available in the literature.